library(forecast)
## Warning: package 'forecast' was built under R version 4.3.3
## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zoo
library(readxl)
library(TTR)
data <- read.csv('aggregated_data.csv')
ts_birth <- ts(data$Records,start=c(1985,1),frequency=12)
plot(ts_birth, main="Birth time series", ylab = "Birth data")
data
## MonthYear Records
## 1 1-1985 300955
## 2 2-1985 278130
## 3 3-1985 307665
## 4 4-1985 298732
## 5 5-1985 314496
## 6 6-1985 305428
## 7 7-1985 330560
## 8 8-1985 333851
## 9 9-1985 328419
## 10 10-1985 320085
## 11 11-1985 298341
## 12 12-1985 307827
## 13 1-1986 303023
## 14 2-1986 279094
## 15 3-1986 309253
## 16 4-1986 300861
## 17 5-1986 312877
## 18 6-1986 304425
## 19 7-1986 331646
## 20 8-1986 331023
## 21 9-1986 331313
## 22 10-1986 315979
## 23 11-1986 289749
## 24 12-1986 309551
## 25 1-1987 301771
## 26 2-1987 280001
## 27 3-1987 314156
## 28 4-1987 304190
## 29 5-1987 316477
## 30 6-1987 317857
## 31 7-1987 333279
## 32 8-1987 328299
## 33 9-1987 331017
## 34 10-1987 323163
## 35 11-1987 302905
## 36 12-1987 317024
## 37 1-1988 306881
## 38 2-1988 294790
## 39 3-1988 318630
## 40 4-1988 305777
## 41 5-1988 322442
## 42 6-1988 325613
## 43 7-1988 343503
## 44 8-1988 351203
## 45 9-1988 343779
## 46 10-1988 328108
## 47 11-1988 311044
## 48 12-1988 318654
## 49 1-1989 320422
## 50 2-1989 300391
## 51 3-1989 339912
## 52 4-1989 318779
## 53 5-1989 336320
## 54 6-1989 338973
## 55 7-1989 356716
## 56 8-1989 366579
## 57 9-1989 357344
## 58 10-1989 344161
## 59 11-1989 325543
## 60 12-1989 335818
## 61 1-1990 335274
## 62 2-1990 312611
## 63 3-1990 350614
## 64 4-1990 336382
## 65 5-1990 354114
## 66 6-1990 347355
## 67 7-1990 367670
## 68 8-1990 372516
## 69 9-1990 358682
## 70 10-1990 353166
## 71 11-1990 333146
## 72 12-1990 336682
## 73 1-1991 335172
## 74 2-1991 309130
## 75 3-1991 344079
## 76 4-1991 335626
## 77 5-1991 353131
## 78 6-1991 334265
## 79 7-1991 362913
## 80 8-1991 366786
## 81 9-1991 356016
## 82 10-1991 348934
## 83 11-1991 323635
## 84 12-1991 341220
## 85 1-1992 334045
## 86 2-1992 315448
## 87 3-1992 339518
## 88 4-1992 333373
## 89 5-1992 344137
## 90 6-1992 339664
## 91 7-1992 359112
## 92 8-1992 348949
## 93 9-1992 347547
## 94 10-1992 343546
## 95 11-1992 321943
## 96 12-1992 337732
## 97 1-1993 323073
## 98 2-1993 304656
## 99 3-1993 342187
## 100 4-1993 327042
## 101 5-1993 335989
## 102 6-1993 335349
## 103 7-1993 352554
## 104 8-1993 350898
## 105 9-1993 348013
## 106 10-1993 332937
## 107 11-1993 316379
## 108 12-1993 331163
## 109 1-1994 320700
## 110 2-1994 301325
## 111 3-1994 339734
## 112 4-1994 317387
## 113 5-1994 330294
## 114 6-1994 329736
## 115 7-1994 345859
## 116 8-1994 352171
## 117 9-1994 339219
## 118 10-1994 330168
## 119 11-1994 319395
## 120 12-1994 326743
## 121 1-1995 316011
## 122 2-1995 295094
## 123 3-1995 328502
## 124 4-1995 309117
## 125 5-1995 334541
## 126 6-1995 329800
## 127 7-1995 340872
## 128 8-1995 350734
## 129 9-1995 339100
## 130 10-1995 330011
## 131 11-1995 310815
## 132 12-1995 314969
## 133 1-1996 314283
## 134 2-1996 301763
## 135 3-1996 322581
## 136 4-1996 312595
## 137 5-1996 325708
## 138 6-1996 318525
## 139 7-1996 345162
## 140 8-1996 346317
## 141 9-1996 336348
## 142 10-1996 336346
## 143 11-1996 309397
## 144 12-1996 322469
## 145 1-1997 317211
## 146 2-1997 291541
## 147 3-1997 321212
## 148 4-1997 314230
## 149 5-1997 330331
## 150 6-1997 321867
## 151 7-1997 346506
## 152 8-1997 339122
## 153 9-1997 333600
## 154 10-1997 328657
## 155 11-1997 307282
## 156 12-1997 329335
## 157 1-1998 319340
## 158 2-1998 298711
## 159 3-1998 329436
## 160 4-1998 319758
## 161 5-1998 330519
## 162 6-1998 327091
## 163 7-1998 348651
## 164 8-1998 344736
## 165 9-1998 343384
## 166 10-1998 332790
## 167 11-1998 313241
## 168 12-1998 333896
## 169 1-1999 319182
## 170 2-1999 297568
## 171 3-1999 332939
## 172 4-1999 316889
## 173 5-1999 328526
## 174 6-1999 332201
## 175 7-1999 349812
## 176 8-1999 351371
## 177 9-1999 349409
## 178 10-1999 332980
## 179 11-1999 315289
## 180 12-1999 333251
## 181 1-2000 330108
## 182 2-2000 317377
## 183 3-2000 340553
## 184 4-2000 317180
## 185 5-2000 341207
## 186 6-2000 341206
## 187 7-2000 348975
## 188 8-2000 360080
## 189 9-2000 347609
## 190 10-2000 343921
## 191 11-2000 333811
## 192 12-2000 336787
## 193 1-2001 335198
## 194 2-2001 303534
## 195 3-2001 338684
## 196 4-2001 323613
## 197 5-2001 344017
## 198 6-2001 331085
## 199 7-2001 351047
## 200 8-2001 361802
## 201 9-2001 342564
## 202 10-2001 344074
## 203 11-2001 323746
## 204 12-2001 326569
## 205 1-2002 330674
## 206 2-2002 303977
## 207 3-2002 331505
## 208 4-2002 324432
## 209 5-2002 339007
## 210 6-2002 327588
## 211 7-2002 357669
## 212 8-2002 359417
## 213 9-2002 348814
## 214 10-2002 345814
## 215 11-2002 318573
## 216 12-2002 334256
## 217 1-2003 315731
## 218 2-2003 294026
## 219 3-2003 322033
## 220 4-2003 315525
## 221 5-2003 331114
## 222 6-2003 322097
## 223 7-2003 347943
## 224 8-2003 344573
## 225 9-2003 344243
## 226 10-2003 338733
## 227 11-2003 306617
## 228 12-2003 329151
## 229 1-2004 318661
## 230 2-2004 301853
## 231 3-2004 330772
## 232 4-2004 318200
## 233 5-2004 322256
## 234 6-2004 329332
## 235 7-2004 343675
## 236 8-2004 340150
## 237 9-2004 340857
## 238 10-2004 333713
## 239 11-2004 321667
## 240 12-2004 331336
## 241 1-2005 317318
## 242 2-2005 295998
## 243 3-2005 333565
## 244 4-2005 317561
## 245 5-2005 330399
## 246 6-2005 335206
## 247 7-2005 341355
## 248 8-2005 353228
## 249 9-2005 347697
## 250 10-2005 329836
## 251 11-2005 321105
## 252 12-2005 333319
## 253 1-2006 325698
## 254 2-2006 305472
## 255 3-2006 341008
## 256 4-2006 314750
## 257 5-2006 338949
## 258 6-2006 341884
## 259 7-2006 351612
## 260 8-2006 370881
## 261 9-2006 358503
## 262 10-2006 351308
## 263 11-2006 336795
## 264 12-2006 340876
## 265 1-2007 339498
## 266 2-2007 312544
## 267 3-2007 344586
## 268 4-2007 322728
## 269 5-2007 345489
## 270 6-2007 342184
## 271 7-2007 362651
## 272 8-2007 373161
## 273 9-2007 350856
## 274 10-2007 353187
## 275 11-2007 338699
## 276 12-2007 339001
## 277 1-2008 340881
## 278 2-2008 323131
## 279 3-2008 334464
## 280 4-2008 330277
## 281 5-2008 338343
## 282 6-2008 332382
## 283 7-2008 358201
## 284 8-2008 356496
## 285 9-2008 351705
## 286 10-2008 341755
## 287 11-2008 309489
## 288 12-2008 338449
## 289 1-2009 322970
## 290 2-2009 302399
## 291 3-2009 331405
## 292 4-2009 321409
## 293 5-2009 328731
## 294 6-2009 330637
## 295 7-2009 351254
## 296 8-2009 343063
## 297 9-2009 345608
## 298 10-2009 331806
## 299 11-2009 305917
## 300 12-2009 326161
## 301 1-2010 308846
## 302 2-2010 288355
## 303 3-2010 322930
## 304 4-2010 309541
## 305 5-2010 312567
## 306 6-2010 318820
## 307 7-2010 329198
## 308 8-2010 333829
## 309 9-2010 334714
## 310 10-2010 321907
## 311 11-2010 311773
## 312 12-2010 323854
## 313 1-2011 306090
## 314 2-2011 284436
## 315 3-2011 314715
## 316 4-2011 298431
## 317 5-2011 311028
## 318 6-2011 321093
## 319 7-2011 329242
## 320 8-2011 342850
## 321 9-2011 329598
## 322 10-2011 313550
## 323 11-2011 307088
## 324 12-2011 312567
## 325 1-2012 301951
## 326 2-2012 290568
## 327 3-2012 308944
## 328 4-2012 292182
## 329 5-2012 314235
## 330 6-2012 311706
## 331 7-2012 331255
## 332 8-2012 344369
## 333 9-2012 324607
## 334 10-2012 329634
## 335 11-2012 310492
## 336 12-2012 309741
## 337 1-2013 308994
## 338 2-2013 278307
## 339 3-2013 305462
## 340 4-2013 297011
## 341 5-2013 313635
## 342 6-2013 304497
## 343 7-2013 332739
## 344 8-2013 337261
## 345 9-2013 322855
## 346 10-2013 324983
## 347 11-2013 304216
## 348 12-2013 320526
## 349 1-2014 311524
## 350 2-2014 284520
## 351 3-2014 308546
## 352 4-2014 303969
## 353 5-2014 318680
## 354 6-2014 310606
## 355 7-2014 339057
## 356 8-2014 337677
## 357 9-2014 332602
## 358 10-2014 327849
## 359 11-2014 303948
## 360 12-2014 325355
## 361 1-2015 311182
## 362 2-2015 284470
## 363 3-2015 313665
## 364 4-2015 305815
## 365 5-2015 312181
## 366 6-2015 314757
## 367 7-2015 336869
## 368 8-2015 335885
## 369 9-2015 331542
## 370 10-2015 323570
## 371 11-2015 304502
## 372 12-2015 320458
Acf(ts_birth, lag.max=48, main = "Birth data ACF")
fivenum(ts_birth)
## [1] 278130.0 314605.5 329517.0 339823.0 373161.0
mean(ts_birth)
## [1] 327413.5
birth_mean <- mean(ts_birth)
birth_median <- median(ts_birth)
boxplot(ts_birth, main="US birth data box plot",ylab="Birth records")
hist(ts_birth, main="Birth data histogram",xlab="Birth records")
abline(v = birth_mean, col = "red", lwd = 1, lty = 1)
abline(v = birth_median, col = "blue", lwd = 1, lty = 1)
decomp <- decompose(ts_birth)
plot(decomp)
attributes(decomp)
## $names
## [1] "x" "seasonal" "trend" "random" "figure" "type"
##
## $class
## [1] "decomposed.ts"
decomp$trend
## Jan Feb Mar Apr May Jun Jul Aug
## 1985 NA NA NA NA NA NA 310460.3 310586.6
## 1986 310763.2 310690.7 310693.4 310642.9 310113.8 309827.7 309847.3 309833.0
## 1987 312044.1 311998.7 311872.8 312159.8 313007.3 313866.9 314391.2 315220.3
## 1988 317911.0 319291.3 320777.4 321515.2 322060.4 322467.4 323099.5 323897.1
## 1989 329807.9 330999.1 332205.0 333439.0 334712.0 336031.3 337365.3 338493.3
## 1990 343999.0 344702.8 345005.9 345436.9 346128.9 346481.7 346513.4 346364.1
## 1991 344240.5 343803.6 343453.8 343166.3 342593.7 342386.5 342528.6 342744.9
## 1992 341982.4 341080.8 339984.7 339407.3 339112.3 338896.5 338294.0 337387.2
## 1993 335320.5 335128.5 335229.1 334806.5 334132.6 333627.0 333254.5 333016.8
## 1994 330647.7 330421.8 330108.4 329626.6 329636.9 329578.4 329198.9 328743.9
## 1995 327010.5 326742.9 326678.0 326666.5 326302.5 325454.4 324891.8 325097.7
## 1996 323675.1 323669.8 323371.1 323520.4 323725.3 323978.7 324413.2 324109.2
## 1997 324425.2 324181.5 323767.2 323332.3 322923.8 323121.8 323496.5 323884.0
## 1998 325869.1 326192.4 326834.0 327413.9 327834.4 328272.7 328456.2 328402.0
## 1999 328715.3 329040.1 329567.6 329826.6 329919.8 329978.3 330406.7 331687.3
## 2000 334943.7 335271.7 335559.6 335940.5 337168.1 338087.2 338446.6 338081.9
## 2001 337362.5 337520.6 337382.1 337178.3 336765.3 335920.2 335305.9 335135.9
## 2002 334191.3 334367.9 334528.9 334861.8 334718.8 334823.5 334521.2 333484.0
## 2003 330017.2 328993.4 328184.5 327699.0 326905.8 326194.9 326104.2 326552.5
## 2004 327516.7 327154.5 326829.2 326478.9 326896.8 327615.0 327650.0 327350.1
## 2005 328357.1 328805.3 329635.2 329758.7 329573.8 329633.0 330064.7 330808.7
## 2006 333285.8 334448.7 335634.5 336979.4 338527.8 339496.5 340386.3 341256.0
## 2007 343543.6 344098.6 343875.0 343634.6 343792.2 343793.5 343773.0 344271.7
## 2008 342900.7 342020.9 341361.9 340920.9 339227.5 337987.4 337218.1 335608.0
## 2009 332514.4 331665.2 330851.5 330182.9 329619.5 328958.7 327858.2 326684.5
## 2010 321153.3 319849.6 319010.9 318144.5 317976.1 318124.0 317913.0 317634.9
## 2011 315924.2 316301.9 316464.6 315903.2 315359.8 314694.3 314051.5 314134.6
## 2012 312957.3 313104.5 312959.8 313422.0 314234.0 314258.1 314433.8 314216.4
## 2013 313228.8 312994.5 312625.3 312358.5 311903.2 312091.1 312645.9 313010.2
## 2014 315298.7 315579.2 316002.7 316528.2 316636.5 316826.5 317013.5 316997.2
## 2015 317288.7 317122.8 317004.0 316781.5 316626.3 316445.4 NA NA
## Sep Oct Nov Dec
## 1985 310692.9 310847.8 310869.0 310759.8
## 1986 310075.0 310418.0 310706.8 311416.4
## 1987 316022.9 316275.5 316590.1 317161.8
## 1988 325017.2 326445.8 327565.8 328700.7
## 1989 339448.4 340627.8 342102.7 343193.3
## 1990 345946.8 345643.0 345570.5 344984.2
## 1991 342818.1 342534.2 342065.6 341915.8
## 1992 337048.7 336896.1 336292.8 335773.5
## 1993 332775.8 332271.3 331631.7 331160.5
## 1994 328016.2 327203.7 327036.0 327215.7
## 1995 325128.9 325027.1 324804.0 323966.1
## 1996 323626.3 323637.4 323898.1 324230.0
## 1997 324525.4 325098.4 325336.6 325562.1
## 1998 328500.3 328526.7 328324.1 328454.0
## 1999 332829.9 333159.3 333699.8 334603.4
## 2000 337427.2 337617.4 338002.5 337697.9
## 2001 334855.2 334590.2 334415.6 334061.1
## 2002 332674.7 331908.9 331208.9 330651.2
## 2003 327242.7 327718.3 327460.7 327393.0
## 2004 327222.5 327312.3 327625.0 328209.0
## 2005 331513.5 331706.5 331945.7 332580.2
## 2006 341699.8 342181.2 342786.2 343071.2
## 2007 344291.1 344183.9 344200.7 343494.5
## 2008 334616.7 334119.8 333349.8 332876.5
## 2009 325746.2 324898.6 323730.6 322564.7
## 2010 317129.3 316324.1 315797.0 315827.6
## 2011 314149.6 313648.8 313522.0 313264.5
## 2012 313560.4 313616.5 313792.8 313467.4
## 2013 313397.6 313816.0 314316.1 314780.9
## 2014 317208.4 317498.6 317304.7 317206.9
## 2015 NA NA NA NA
decomp$seasonal
## Jan Feb Mar Apr May Jun
## 1985 -8176.5471 -29475.5346 600.6362 -12440.5416 1966.5279 -707.6721
## 1986 -8176.5471 -29475.5346 600.6362 -12440.5416 1966.5279 -707.6721
## 1987 -8176.5471 -29475.5346 600.6362 -12440.5416 1966.5279 -707.6721
## 1988 -8176.5471 -29475.5346 600.6362 -12440.5416 1966.5279 -707.6721
## 1989 -8176.5471 -29475.5346 600.6362 -12440.5416 1966.5279 -707.6721
## 1990 -8176.5471 -29475.5346 600.6362 -12440.5416 1966.5279 -707.6721
## 1991 -8176.5471 -29475.5346 600.6362 -12440.5416 1966.5279 -707.6721
## 1992 -8176.5471 -29475.5346 600.6362 -12440.5416 1966.5279 -707.6721
## 1993 -8176.5471 -29475.5346 600.6362 -12440.5416 1966.5279 -707.6721
## 1994 -8176.5471 -29475.5346 600.6362 -12440.5416 1966.5279 -707.6721
## 1995 -8176.5471 -29475.5346 600.6362 -12440.5416 1966.5279 -707.6721
## 1996 -8176.5471 -29475.5346 600.6362 -12440.5416 1966.5279 -707.6721
## 1997 -8176.5471 -29475.5346 600.6362 -12440.5416 1966.5279 -707.6721
## 1998 -8176.5471 -29475.5346 600.6362 -12440.5416 1966.5279 -707.6721
## 1999 -8176.5471 -29475.5346 600.6362 -12440.5416 1966.5279 -707.6721
## 2000 -8176.5471 -29475.5346 600.6362 -12440.5416 1966.5279 -707.6721
## 2001 -8176.5471 -29475.5346 600.6362 -12440.5416 1966.5279 -707.6721
## 2002 -8176.5471 -29475.5346 600.6362 -12440.5416 1966.5279 -707.6721
## 2003 -8176.5471 -29475.5346 600.6362 -12440.5416 1966.5279 -707.6721
## 2004 -8176.5471 -29475.5346 600.6362 -12440.5416 1966.5279 -707.6721
## 2005 -8176.5471 -29475.5346 600.6362 -12440.5416 1966.5279 -707.6721
## 2006 -8176.5471 -29475.5346 600.6362 -12440.5416 1966.5279 -707.6721
## 2007 -8176.5471 -29475.5346 600.6362 -12440.5416 1966.5279 -707.6721
## 2008 -8176.5471 -29475.5346 600.6362 -12440.5416 1966.5279 -707.6721
## 2009 -8176.5471 -29475.5346 600.6362 -12440.5416 1966.5279 -707.6721
## 2010 -8176.5471 -29475.5346 600.6362 -12440.5416 1966.5279 -707.6721
## 2011 -8176.5471 -29475.5346 600.6362 -12440.5416 1966.5279 -707.6721
## 2012 -8176.5471 -29475.5346 600.6362 -12440.5416 1966.5279 -707.6721
## 2013 -8176.5471 -29475.5346 600.6362 -12440.5416 1966.5279 -707.6721
## 2014 -8176.5471 -29475.5346 600.6362 -12440.5416 1966.5279 -707.6721
## 2015 -8176.5471 -29475.5346 600.6362 -12440.5416 1966.5279 -707.6721
## Jul Aug Sep Oct Nov Dec
## 1985 18552.3890 21952.8459 14356.7071 6574.5029 -12800.8193 -402.4943
## 1986 18552.3890 21952.8459 14356.7071 6574.5029 -12800.8193 -402.4943
## 1987 18552.3890 21952.8459 14356.7071 6574.5029 -12800.8193 -402.4943
## 1988 18552.3890 21952.8459 14356.7071 6574.5029 -12800.8193 -402.4943
## 1989 18552.3890 21952.8459 14356.7071 6574.5029 -12800.8193 -402.4943
## 1990 18552.3890 21952.8459 14356.7071 6574.5029 -12800.8193 -402.4943
## 1991 18552.3890 21952.8459 14356.7071 6574.5029 -12800.8193 -402.4943
## 1992 18552.3890 21952.8459 14356.7071 6574.5029 -12800.8193 -402.4943
## 1993 18552.3890 21952.8459 14356.7071 6574.5029 -12800.8193 -402.4943
## 1994 18552.3890 21952.8459 14356.7071 6574.5029 -12800.8193 -402.4943
## 1995 18552.3890 21952.8459 14356.7071 6574.5029 -12800.8193 -402.4943
## 1996 18552.3890 21952.8459 14356.7071 6574.5029 -12800.8193 -402.4943
## 1997 18552.3890 21952.8459 14356.7071 6574.5029 -12800.8193 -402.4943
## 1998 18552.3890 21952.8459 14356.7071 6574.5029 -12800.8193 -402.4943
## 1999 18552.3890 21952.8459 14356.7071 6574.5029 -12800.8193 -402.4943
## 2000 18552.3890 21952.8459 14356.7071 6574.5029 -12800.8193 -402.4943
## 2001 18552.3890 21952.8459 14356.7071 6574.5029 -12800.8193 -402.4943
## 2002 18552.3890 21952.8459 14356.7071 6574.5029 -12800.8193 -402.4943
## 2003 18552.3890 21952.8459 14356.7071 6574.5029 -12800.8193 -402.4943
## 2004 18552.3890 21952.8459 14356.7071 6574.5029 -12800.8193 -402.4943
## 2005 18552.3890 21952.8459 14356.7071 6574.5029 -12800.8193 -402.4943
## 2006 18552.3890 21952.8459 14356.7071 6574.5029 -12800.8193 -402.4943
## 2007 18552.3890 21952.8459 14356.7071 6574.5029 -12800.8193 -402.4943
## 2008 18552.3890 21952.8459 14356.7071 6574.5029 -12800.8193 -402.4943
## 2009 18552.3890 21952.8459 14356.7071 6574.5029 -12800.8193 -402.4943
## 2010 18552.3890 21952.8459 14356.7071 6574.5029 -12800.8193 -402.4943
## 2011 18552.3890 21952.8459 14356.7071 6574.5029 -12800.8193 -402.4943
## 2012 18552.3890 21952.8459 14356.7071 6574.5029 -12800.8193 -402.4943
## 2013 18552.3890 21952.8459 14356.7071 6574.5029 -12800.8193 -402.4943
## 2014 18552.3890 21952.8459 14356.7071 6574.5029 -12800.8193 -402.4943
## 2015 18552.3890 21952.8459 14356.7071 6574.5029 -12800.8193 -402.4943
decomp$type
## [1] "additive"
s_adj <- seasadj(decomp)
plot(ts_birth,
main = 'Seasonally adjusted time series',
xlab = 'Time',
ylab = 'Birth records')
lines(s_adj, col='red')
naive_f <- naive(ts_birth,60)
plot(naive_f$residuals, xlab="Time", ylab="Residuals")
hist(naive_f$residuals, xlab="Residuals", main="Residuals histogram")
naive_f$fitted
## Jan Feb Mar Apr May Jun Jul Aug Sep Oct
## 1985 NA 300955 278130 307665 298732 314496 305428 330560 333851 328419
## 1986 307827 303023 279094 309253 300861 312877 304425 331646 331023 331313
## 1987 309551 301771 280001 314156 304190 316477 317857 333279 328299 331017
## 1988 317024 306881 294790 318630 305777 322442 325613 343503 351203 343779
## 1989 318654 320422 300391 339912 318779 336320 338973 356716 366579 357344
## 1990 335818 335274 312611 350614 336382 354114 347355 367670 372516 358682
## 1991 336682 335172 309130 344079 335626 353131 334265 362913 366786 356016
## 1992 341220 334045 315448 339518 333373 344137 339664 359112 348949 347547
## 1993 337732 323073 304656 342187 327042 335989 335349 352554 350898 348013
## 1994 331163 320700 301325 339734 317387 330294 329736 345859 352171 339219
## 1995 326743 316011 295094 328502 309117 334541 329800 340872 350734 339100
## 1996 314969 314283 301763 322581 312595 325708 318525 345162 346317 336348
## 1997 322469 317211 291541 321212 314230 330331 321867 346506 339122 333600
## 1998 329335 319340 298711 329436 319758 330519 327091 348651 344736 343384
## 1999 333896 319182 297568 332939 316889 328526 332201 349812 351371 349409
## 2000 333251 330108 317377 340553 317180 341207 341206 348975 360080 347609
## 2001 336787 335198 303534 338684 323613 344017 331085 351047 361802 342564
## 2002 326569 330674 303977 331505 324432 339007 327588 357669 359417 348814
## 2003 334256 315731 294026 322033 315525 331114 322097 347943 344573 344243
## 2004 329151 318661 301853 330772 318200 322256 329332 343675 340150 340857
## 2005 331336 317318 295998 333565 317561 330399 335206 341355 353228 347697
## 2006 333319 325698 305472 341008 314750 338949 341884 351612 370881 358503
## 2007 340876 339498 312544 344586 322728 345489 342184 362651 373161 350856
## 2008 339001 340881 323131 334464 330277 338343 332382 358201 356496 351705
## 2009 338449 322970 302399 331405 321409 328731 330637 351254 343063 345608
## 2010 326161 308846 288355 322930 309541 312567 318820 329198 333829 334714
## 2011 323854 306090 284436 314715 298431 311028 321093 329242 342850 329598
## 2012 312567 301951 290568 308944 292182 314235 311706 331255 344369 324607
## 2013 309741 308994 278307 305462 297011 313635 304497 332739 337261 322855
## 2014 320526 311524 284520 308546 303969 318680 310606 339057 337677 332602
## 2015 325355 311182 284470 313665 305815 312181 314757 336869 335885 331542
## Nov Dec
## 1985 320085 298341
## 1986 315979 289749
## 1987 323163 302905
## 1988 328108 311044
## 1989 344161 325543
## 1990 353166 333146
## 1991 348934 323635
## 1992 343546 321943
## 1993 332937 316379
## 1994 330168 319395
## 1995 330011 310815
## 1996 336346 309397
## 1997 328657 307282
## 1998 332790 313241
## 1999 332980 315289
## 2000 343921 333811
## 2001 344074 323746
## 2002 345814 318573
## 2003 338733 306617
## 2004 333713 321667
## 2005 329836 321105
## 2006 351308 336795
## 2007 353187 338699
## 2008 341755 309489
## 2009 331806 305917
## 2010 321907 311773
## 2011 313550 307088
## 2012 329634 310492
## 2013 324983 304216
## 2014 327849 303948
## 2015 323570 304502
naive_f$residuals
## Jan Feb Mar Apr May Jun Jul Aug Sep Oct
## 1985 NA -22825 29535 -8933 15764 -9068 25132 3291 -5432 -8334
## 1986 -4804 -23929 30159 -8392 12016 -8452 27221 -623 290 -15334
## 1987 -7780 -21770 34155 -9966 12287 1380 15422 -4980 2718 -7854
## 1988 -10143 -12091 23840 -12853 16665 3171 17890 7700 -7424 -15671
## 1989 1768 -20031 39521 -21133 17541 2653 17743 9863 -9235 -13183
## 1990 -544 -22663 38003 -14232 17732 -6759 20315 4846 -13834 -5516
## 1991 -1510 -26042 34949 -8453 17505 -18866 28648 3873 -10770 -7082
## 1992 -7175 -18597 24070 -6145 10764 -4473 19448 -10163 -1402 -4001
## 1993 -14659 -18417 37531 -15145 8947 -640 17205 -1656 -2885 -15076
## 1994 -10463 -19375 38409 -22347 12907 -558 16123 6312 -12952 -9051
## 1995 -10732 -20917 33408 -19385 25424 -4741 11072 9862 -11634 -9089
## 1996 -686 -12520 20818 -9986 13113 -7183 26637 1155 -9969 -2
## 1997 -5258 -25670 29671 -6982 16101 -8464 24639 -7384 -5522 -4943
## 1998 -9995 -20629 30725 -9678 10761 -3428 21560 -3915 -1352 -10594
## 1999 -14714 -21614 35371 -16050 11637 3675 17611 1559 -1962 -16429
## 2000 -3143 -12731 23176 -23373 24027 -1 7769 11105 -12471 -3688
## 2001 -1589 -31664 35150 -15071 20404 -12932 19962 10755 -19238 1510
## 2002 4105 -26697 27528 -7073 14575 -11419 30081 1748 -10603 -3000
## 2003 -18525 -21705 28007 -6508 15589 -9017 25846 -3370 -330 -5510
## 2004 -10490 -16808 28919 -12572 4056 7076 14343 -3525 707 -7144
## 2005 -14018 -21320 37567 -16004 12838 4807 6149 11873 -5531 -17861
## 2006 -7621 -20226 35536 -26258 24199 2935 9728 19269 -12378 -7195
## 2007 -1378 -26954 32042 -21858 22761 -3305 20467 10510 -22305 2331
## 2008 1880 -17750 11333 -4187 8066 -5961 25819 -1705 -4791 -9950
## 2009 -15479 -20571 29006 -9996 7322 1906 20617 -8191 2545 -13802
## 2010 -17315 -20491 34575 -13389 3026 6253 10378 4631 885 -12807
## 2011 -17764 -21654 30279 -16284 12597 10065 8149 13608 -13252 -16048
## 2012 -10616 -11383 18376 -16762 22053 -2529 19549 13114 -19762 5027
## 2013 -747 -30687 27155 -8451 16624 -9138 28242 4522 -14406 2128
## 2014 -9002 -27004 24026 -4577 14711 -8074 28451 -1380 -5075 -4753
## 2015 -14173 -26712 29195 -7850 6366 2576 22112 -984 -4343 -7972
## Nov Dec
## 1985 -21744 9486
## 1986 -26230 19802
## 1987 -20258 14119
## 1988 -17064 7610
## 1989 -18618 10275
## 1990 -20020 3536
## 1991 -25299 17585
## 1992 -21603 15789
## 1993 -16558 14784
## 1994 -10773 7348
## 1995 -19196 4154
## 1996 -26949 13072
## 1997 -21375 22053
## 1998 -19549 20655
## 1999 -17691 17962
## 2000 -10110 2976
## 2001 -20328 2823
## 2002 -27241 15683
## 2003 -32116 22534
## 2004 -12046 9669
## 2005 -8731 12214
## 2006 -14513 4081
## 2007 -14488 302
## 2008 -32266 28960
## 2009 -25889 20244
## 2010 -10134 12081
## 2011 -6462 5479
## 2012 -19142 -751
## 2013 -20767 16310
## 2014 -23901 21407
## 2015 -19068 15956
plot(naive_f$residuals~naive_f$fitted, main = "Residual-Fitted values plot",ylab="Residuals",xlab="Fitted values")
plot(naive_f$residuals~naive_f$x, main = "Residual-Actual values plot",ylab="Residuals",xlab="Actual values")
Acf (naive_f$residuals, lag.max=48, main="Residuals ACF")
naive_ac <- accuracy(naive_f)
naive_forecast <- forecast(naive_f,h=12)
plot(ts_birth, main = "Birth time series",ylab="Birth records")
lines(naive_forecast$mean, col='red')
plot(naive_forecast)
# Moving average
plot(ts_birth, xlab="Time",ylab="Birth Records", main = "Time series and forecasts")
# I am using the simple moving average
# SMA: n =3, in red
sma_3 <- SMA(ts_birth,n= 3)
lines(sma_3, col='red')
sma3_f <- forecast(sma_3,h=60)
## Warning in ets(object, lambda = lambda, biasadj = biasadj,
## allow.multiplicative.trend = allow.multiplicative.trend, : Missing values
## encountered. Using longest contiguous portion of time series
# SMA: n =6, in blue
sma_6 <- SMA(ts_birth,n= 6)
lines(sma_6, col='blue')
sma6_f <- forecast(sma_6,h=60)
## Warning in ets(object, lambda = lambda, biasadj = biasadj,
## allow.multiplicative.trend = allow.multiplicative.trend, : Missing values
## encountered. Using longest contiguous portion of time series
# SMA: n =12, in green
sma_12 <- SMA(ts_birth,n= 12)
lines(sma_12, col='green')
sma12_f <- forecast(sma_12,h=60)
## Warning in ets(object, lambda = lambda, biasadj = biasadj,
## allow.multiplicative.trend = allow.multiplicative.trend, : Missing values
## encountered. Using longest contiguous portion of time series
plot(sma3_f)
plot(sma6_f)
plot(sma12_f)
plot(sma3_f$residuals, xlab="Time", ylab="Residuals")
hist(sma3_f$residuals, xlab="Residuals", main="Residuals histogram")
plot(sma3_f$residuals~sma3_f$fitted, main = "Residual-Fitted values plot",ylab="Residuals",xlab="Fitted values")
plot(sma3_f$residuals~sma3_f$x, main = "Residual-Actual values plot",ylab="Residuals",xlab="Actual values")
Acf(sma3_f$residuals,lag.max=48,main="Residuals ACF")
sma3_f$residuals
## Jan Feb Mar Apr May
## 1985 -3242.962309 84.541714 1148.549027
## 1986 -830.224583 -896.984390 197.724160 602.666758 679.936901
## 1987 211.408905 2331.208574 1273.491521 2130.863055 1562.747451
## 1988 -540.102493 2834.428190 236.451990 936.798332 -1381.720849
## 1989 2377.320919 2000.556884 6805.520922 704.558159 1336.475276
## 1990 1952.310586 1226.057442 4646.789584 1641.684513 3200.602061
## 1991 -1048.241596 -2409.171639 2265.148674 1517.607744 4109.580790
## 1992 -20.414444 2850.760842 -835.874457 1118.975399 -1008.778113
## 1993 -1902.538737 -180.902374 1250.208048 2671.594740 -138.670752
## 1994 877.559313 562.664866 2613.320086 223.013999 -918.345423
## 1995 207.399730 -2537.047944 360.147387 -932.222855 2617.271798
## 1996 -272.886538 2586.894877 2290.870508 766.553180 -2597.894426
## 1997 -1470.541866 -386.073378 -665.037737 366.998796 2380.760676
## 1998 1792.882350 2667.019943 -277.650067 1442.861884 -3.557737
## 1999 362.319752 314.145251 -593.771710 571.536803 -251.568196
## 2000 3965.609253 6217.772672 2084.895554 -3062.026516 -2661.701512
## 2001 1967.330635 -4588.734600 380.450577 -2516.236102 2961.876181
## 2002 466.828139 -1022.338120 1406.671267 -734.829037 1124.778733
## 2003 -5109.799131 -2570.349841 -4260.731232 1377.883672 1870.569389
## 2004 -1770.891138 3990.345894 259.904038 1168.736808 -3788.701966
## 2005 -598.648748 -3034.341557 490.296678 1424.285989 892.310374
## 2006 3528.633651 302.662504 2267.643203 -2358.799001 580.491587
## 2007 929.978710 -2575.251521 966.335169 -4264.407139 449.834141
## 2008 792.306037 342.996038 -1794.357232 -2189.597466 -5467.597623
## 2009 -1294.428769 3248.781942 -2595.028942 859.967744 -1763.159257
## 2010 -2683.935000 -226.498735 -1274.304723 1637.734599 -2456.551379
## 2011 -394.186512 -3582.456755 -3287.543417 -1161.335602 -1644.594130
## 2012 1068.084833 54.116781 -1462.097908 -1895.082262 -2636.512149
## 2013 -2000.889766 -5180.225428 -1636.628340 -2594.846061 1288.327334
## 2014 365.689446 -1063.787737 -4284.496221 -1156.883746 853.745094
## 2015 -684.703699 -965.119139 -4167.423365 -413.416725 -1292.144850
## Jun Jul Aug Sep Oct
## 1985 -574.429284 -57.326210 -336.577394 2452.311468 341.661492
## 1986 -1119.084727 -144.429790 -529.982244 3920.472157 -1267.865618
## 1987 1702.966732 -755.943640 -2669.178737 -659.321373 628.176188
## 1988 2824.691883 2134.949699 2956.889961 934.419308 -1209.013576
## 1989 122.669971 2183.794583 3438.959527 984.467896 -277.513831
## 1990 -664.717487 -36.721063 -493.803901 -1306.616732 -857.429782
## 1991 -2797.038551 -1307.343842 -2012.052026 2235.196881 -664.665407
## 1992 565.326788 -1823.703857 -4955.236099 -2354.548314 -1118.704417
## 1993 -1781.470211 -1890.053917 -1580.657546 -789.038147 -2508.216918
## 1994 -2819.693084 -880.720634 750.030453 -1867.074071 -1202.881340
## 1995 944.377729 172.743253 -1189.014282 -1943.381306 395.654427
## 1996 -824.990115 474.879778 305.638644 900.360273 1049.632151
## 1997 718.139151 339.175182 -3663.445163 -1113.034773 -1926.515254
## 1998 -302.826732 -794.427252 -1847.080420 393.886811 -1289.643862
## 1999 263.695885 567.914671 1030.200656 669.695575 -1639.588080
## 2000 725.333730 186.188603 -294.635598 -2918.904741 2334.148715
## 2001 -2025.304278 -1240.209510 -620.252740 -1193.211643 1702.654960
## 2002 -793.608608 684.680858 227.253219 2024.857477 16.962646
## 2003 573.680446 429.265993 -2073.126571 2367.985133 925.447543
## 2004 52.343690 -1894.520982 -578.111813 -1174.788016 709.463415
## 2005 1042.198504 -2494.958070 1040.198785 -890.223821 156.812464
## 2006 788.150960 1865.095990 4032.265391 421.180040 3828.914267
## 2007 -265.009824 2927.656880 2635.241276 -2194.532621 828.167195
## 2008 -102.853684 -1028.468284 -462.113746 1449.671306 -1461.668463
## 2009 293.459184 -425.544034 -1772.179831 -17.144825 -2459.211273
## 2010 -803.744917 -3796.339355 592.776806 310.698394 1605.996944
## 2011 2698.713142 -108.371323 4049.109019 -2238.756411 -1238.593049
## 2012 1489.338235 2656.048659 3465.775865 -783.040200 3428.307312
## 2013 236.597335 1544.909567 1311.195603 1070.852775 1391.534899
## 2014 1213.522882 1296.221169 -257.571924 2278.981222 225.206565
## 2015 915.803940 -24.889977 1344.101503 551.456418 -446.281667
## Nov Dec
## 1985 -383.665852 -2045.814792
## 1986 -2192.186035 -2326.531853
## 1987 3118.937701 222.610720
## 1988 -1846.642176 -3471.239866
## 1989 -2160.809540 -2285.872764
## 1990 -1543.048670 -2400.326735
## 1991 -2791.823978 23.000279
## 1992 2655.533822 1679.647846
## 1993 133.176299 -653.014814
## 1994 697.770987 786.275663
## 1995 -1708.925675 -3094.872415
## 1996 -738.099824 289.271688
## 1997 1014.299539 3523.620044
## 1998 1101.183969 1759.058287
## 1999 -445.972943 -462.715455
## 2000 2823.030545 1280.194741
## 2001 -1092.775845 -393.145651
## 2002 -2051.512127 72.024597
## 2003 -1076.664820 -107.043644
## 2004 5445.295688 1707.794762
## 2005 876.726242 119.304873
## 2006 131.639111 -1029.367167
## 2007 80.727584 955.117629
## 2008 -4047.926933 572.961928
## 2009 -747.847495 -1514.809701
## 2010 4250.269347 1270.391139
## 2011 -324.431073 -743.195444
## 2012 237.163190 -81.595182
## 2013 542.526432 4116.830036
## 2014 313.790517 2478.801160
## 2015 1123.178345 1213.260416
plot(sma6_f$residuals, xlab="Time", ylab="Residuals")
hist(sma6_f$residuals, xlab="Residuals", main="Residuals histogram")
plot(sma6_f$residuals~sma6_f$fitted, main = "Residual-Fitted values plot",xlab="Residuals",ylab="Fitted values")
plot(sma6_f$residuals~sma6_f$x, main = "Residual-Actual values plot",xlab="Residuals",ylab="Actual values")
attributes(sma6_f)
## $names
## [1] "model" "mean" "level" "x" "upper" "lower"
## [7] "fitted" "method" "series" "residuals"
##
## $class
## [1] "forecast"
sma6_f$residuals
## Jan Feb Mar Apr May
## 1985
## 1986 -1.063278e-03 -2.244091e-03 -3.080652e-03 1.414281e-04 7.638412e-05
## 1987 -1.648231e-03 4.387300e-04 -1.735079e-03 4.352518e-03 6.366863e-03
## 1988 -9.598788e-05 9.416404e-03 -5.399218e-04 -1.664365e-04 1.687900e-03
## 1989 1.324331e-03 -3.006865e-04 4.681111e-03 4.116022e-03 4.156820e-03
## 1990 1.550108e-03 -2.334662e-03 2.860855e-03 4.751801e-03 5.548347e-03
## 1991 -2.238602e-03 -5.148907e-03 8.137092e-04 1.712879e-03 3.231739e-03
## 1992 -6.560075e-04 4.551063e-04 -1.063113e-03 2.125580e-03 2.948980e-03
## 1993 -3.423232e-03 5.063436e-03 4.932321e-03 1.273457e-03 -5.209127e-04
## 1994 -1.421256e-03 2.127127e-03 3.601105e-03 2.045113e-03 -4.656590e-04
## 1995 -9.608935e-04 -2.273315e-03 2.424937e-03 -6.375836e-04 6.444170e-04
## 1996 1.323791e-03 2.236053e-03 -8.953621e-04 1.441667e-03 4.031640e-04
## 1997 -8.647642e-04 -1.859927e-03 -6.045595e-04 -1.478403e-03 3.579842e-03
## 1998 -8.150903e-05 5.915418e-03 4.472352e-03 4.325401e-03 3.058574e-03
## 1999 -1.563104e-03 2.220082e-03 1.481694e-03 1.247132e-03 -1.993416e-04
## 2000 3.295541e-03 8.312833e-03 1.046784e-03 2.698229e-04 4.455473e-03
## 2001 6.188441e-03 -3.406277e-03 2.088632e-03 -1.198039e-03 -2.621830e-03
## 2002 3.913309e-03 -3.021301e-03 1.919743e-03 -8.138241e-05 4.674980e-04
## 2003 -7.813056e-03 -6.467069e-03 -5.407922e-03 -4.138027e-03 8.307593e-04
## 2004 -1.342313e-03 4.620503e-03 -2.894167e-04 -1.145541e-03 2.860425e-04
## 2005 -9.687661e-05 3.521494e-03 2.741088e-03 6.431270e-04 -3.920339e-03
## 2006 5.495595e-03 1.188368e-03 2.897604e-03 1.059085e-03 6.264889e-04
## 2007 5.519146e-03 -5.317116e-03 -1.147321e-03 -5.558482e-03 -3.061074e-03
## 2008 1.965317e-03 9.673251e-05 -1.812949e-03 -2.176347e-03 -7.438525e-03
## 2009 -3.163367e-03 -6.654899e-05 -2.226466e-03 5.484429e-04 3.301567e-03
## 2010 -6.741358e-03 1.533232e-04 -3.111400e-03 1.928955e-05 -2.828711e-03
## 2011 1.701926e-03 7.253120e-04 -3.727507e-03 -2.325812e-03 -8.022411e-03
## 2012 -5.399620e-05 -2.410260e-04 -3.524410e-03 -6.947532e-04 -3.577207e-03
## 2013 1.339088e-03 -8.813272e-03 -2.770449e-03 -7.018310e-03 -5.182396e-03
## 2014 1.442218e-03 -2.130919e-03 -1.285591e-03 -1.800124e-03 -5.647173e-04
## 2015 -1.844774e-03 -1.830271e-03 -3.320509e-03 -1.731501e-03 -3.399290e-03
## Jun Jul Aug Sep Oct
## 1985 -7.636175e-03 -1.464918e-04 1.296616e-03 2.636724e-03 2.995434e-04
## 1986 -1.259343e-03 8.262241e-04 4.384091e-04 4.404708e-03 -2.215692e-03
## 1987 4.072401e-03 8.433387e-04 -2.730333e-03 9.287137e-04 -4.403607e-04
## 1988 4.097935e-03 3.465657e-03 1.129320e-03 4.362635e-03 2.392921e-04
## 1989 9.327530e-03 1.982729e-03 5.030738e-03 -7.001992e-04 1.261908e-03
## 1990 4.342307e-03 3.394314e-04 2.332990e-03 -4.376044e-03 -1.766314e-03
## 1991 -6.244523e-04 2.717320e-04 3.154847e-03 -1.122249e-03 -2.533694e-03
## 1992 -6.341821e-04 -1.412228e-03 -8.552969e-03 -1.769183e-03 -2.937313e-03
## 1993 -8.662562e-04 9.668307e-04 -2.581623e-03 -3.652241e-03 -5.503879e-03
## 1994 -2.819385e-04 -1.195784e-03 -7.760286e-05 -6.897320e-03 -1.900973e-03
## 1995 2.340243e-03 -1.429208e-03 2.302188e-03 -1.730739e-03 1.247489e-03
## 1996 2.557224e-03 1.606917e-03 -3.699107e-03 1.876533e-04 2.755109e-03
## 1997 -2.537353e-05 7.522562e-04 -2.098811e-03 -6.307756e-04 -1.911343e-03
## 1998 -2.244594e-03 -1.590500e-04 -3.500606e-03 -2.462313e-04 -2.951774e-03
## 1999 -8.316522e-04 1.310713e-03 8.737563e-04 9.246101e-04 -1.634353e-03
## 2000 2.889776e-03 -5.964558e-03 -4.903384e-03 -3.304886e-03 4.305034e-03
## 2001 -2.440545e-03 -5.579466e-03 4.262071e-03 -4.548919e-03 1.672694e-03
## 2002 1.072605e-03 -3.273300e-04 2.025433e-03 1.494608e-03 1.015916e-03
## 2003 -4.318331e-03 4.260602e-03 6.704431e-04 5.059777e-03 2.446100e-03
## 2004 3.462999e-04 -1.527806e-03 -6.532433e-03 -1.100122e-03 -7.293680e-04
## 2005 2.202567e-03 -2.278792e-03 2.802065e-03 -2.636875e-04 -3.397323e-03
## 2006 3.906076e-03 -2.041873e-03 6.092994e-03 4.324503e-04 7.394927e-03
## 2007 1.381326e-03 -2.083185e-03 4.771789e-03 -3.846349e-03 5.845822e-03
## 2008 -1.989104e-03 -4.089243e-03 -7.416883e-03 3.647335e-03 -2.128463e-03
## 2009 -3.126421e-03 1.095929e-03 -4.801422e-03 1.186167e-03 -3.292423e-03
## 2010 -1.870683e-03 -2.259406e-03 -1.462990e-03 4.853700e-04 -1.914135e-03
## 2011 1.792659e-04 -1.372852e-03 4.704885e-03 8.799546e-04 -1.582901e-03
## 2012 9.039829e-04 1.594014e-03 1.667567e-03 1.107215e-03 9.752818e-03
## 2013 -8.091366e-04 -6.335056e-04 5.341525e-03 2.384933e-03 5.045975e-03
## 2014 -5.206901e-03 3.105899e-04 1.193779e-03 5.390512e-03 2.177574e-03
## 2015 -4.790190e-03 -6.722105e-05 8.385792e-04 2.705730e-03 -2.049719e-04
## Nov Dec
## 1985 -1.640006e-03 2.533668e-04
## 1986 -4.380826e-03 2.791122e-03
## 1987 2.114466e-04 -1.197858e-03
## 1988 3.470015e-04 -5.035166e-03
## 1989 1.459055e-04 -3.401915e-03
## 1990 -3.411631e-03 -5.408515e-03
## 1991 -6.653814e-03 4.027342e-03
## 1992 -2.215385e-03 1.356509e-04
## 1993 -1.086006e-03 -1.198482e-03
## 1994 3.069026e-03 -1.237176e-03
## 1995 -4.417523e-03 -7.247979e-03
## 1996 -8.481169e-04 1.629411e-03
## 1997 -3.713586e-03 4.218633e-03
## 1998 -1.001243e-03 3.331462e-03
## 1999 6.758714e-04 -6.393291e-06
## 2000 3.476996e-03 -2.971306e-03
## 2001 -2.112988e-03 -1.845497e-03
## 2002 -3.057396e-03 2.951309e-03
## 2003 -4.880679e-03 3.659144e-03
## 2004 7.985433e-03 4.420199e-04
## 2005 2.926576e-03 -1.456795e-03
## 2006 4.232220e-03 -2.834265e-03
## 2007 3.388884e-03 -2.580092e-03
## 2008 -5.378874e-03 4.474442e-03
## 2009 -2.884441e-03 -1.321397e-03
## 2010 8.650034e-03 2.408784e-03
## 2011 6.055698e-03 -5.087494e-03
## 2012 4.657916e-03 -2.796922e-03
## 2013 2.446209e-03 7.346906e-03
## 2014 -5.870696e-04 6.707683e-03
## 2015 3.966859e-03 2.344506e-03
plot(sma12_f$residuals, xlab="Time", ylab="Residuals")
hist(sma12_f$residuals, xlab="Residuals", main="Residuals histogram")
plot(sma12_f$residuals~sma12_f$fitted, main = "Residual-Fitted values plot",xlab="Residuals",ylab="Fitted values")
plot(sma12_f$residuals~sma12_f$x, main = "Residual-Actual values plot",xlab="Residuals",ylab="Actual values")
ses <- HoltWinters(ts_birth,beta = FALSE,gamma=FALSE)
ses_forecast <- forecast(ses,h=60)
ses$alpha
## [1] 0.5762466
ses_forecast$fitted[2]
## [1] 300955
sd(ses_forecast$residuals)
## [1] NA
plot(ses_forecast$residuals, xlab="Time", ylab="Residuals")
hist(ses_forecast$residuals, xlab="Residuals", main="Residuals histogram")
Acf (ses_forecast$residuals,lag.max=48,main="Residuals ACF")
plot(ses_forecast$residuals~naive_f$fitted, main = "Residual-Fitted values plot",ylab="Residuals",xlab="Fitted values")
plot(ses_forecast$residuals~naive_f$x, main = "Residual-Actual values plot",ylab="Residuals",xlab="Actual values")
plot(ses_forecast, main ="Simple Exponential Smoothing forecast")
plot(ts_birth, main="Actual data & forecasts",ylab="Birth records")
lines(ses_forecast$fitted,col='red')
HW <- HoltWinters(ts_birth)
HW_f <- forecast(HW,h=60)
HW$alpha
## alpha
## 0.2066015
HW$beta
## beta
## 0.1299084
HW$gamma
## gamma
## 0.1355967
plot(ts_birth, main="Actual data & forecasts",ylab="Birth records")
lines(HW_f$fitted,col='red')
plot(HW_f$residuals, xlab="Time", ylab="Residuals")
hist(HW_f$residuals, xlab="Residuals", main="Residuals histogram")
Acf (HW_f$residuals, lag.max=48, main="Residuals ACF")
plot(HW_f$residuals~naive_f$fitted, main = "Residual-Fitted values plot",ylab="Residuals",xlab="Fitted values")
plot(HW_f$residuals~naive_f$x, main = "Residual-Actual values plot",ylab="Residuals",xlab="Actual values")
plot(HW_f)
auto_fit <- auto.arima(ts_birth,trace=TRUE,stepwise=FALSE)
##
## Fitting models using approximations to speed things up...
##
## ARIMA(0,1,0)(0,1,0)[12] : 7224.06
## ARIMA(0,1,0)(0,1,1)[12] : Inf
## ARIMA(0,1,0)(0,1,2)[12] : Inf
## ARIMA(0,1,0)(1,1,0)[12] : 7231.767
## ARIMA(0,1,0)(1,1,1)[12] : 7137.604
## ARIMA(0,1,0)(1,1,2)[12] : Inf
## ARIMA(0,1,0)(2,1,0)[12] : 7204.457
## ARIMA(0,1,0)(2,1,1)[12] : 7040.941
## ARIMA(0,1,0)(2,1,2)[12] : 6998.324
## ARIMA(0,1,1)(0,1,0)[12] : 7067.526
## ARIMA(0,1,1)(0,1,1)[12] : Inf
## ARIMA(0,1,1)(0,1,2)[12] : Inf
## ARIMA(0,1,1)(1,1,0)[12] : 7060.617
## ARIMA(0,1,1)(1,1,1)[12] : 6970.324
## ARIMA(0,1,1)(1,1,2)[12] : Inf
## ARIMA(0,1,1)(2,1,0)[12] : 7050.7
## ARIMA(0,1,1)(2,1,1)[12] : 6943.139
## ARIMA(0,1,1)(2,1,2)[12] : 6934.887
## ARIMA(0,1,2)(0,1,0)[12] : 7069.541
## ARIMA(0,1,2)(0,1,1)[12] : 6953.398
## ARIMA(0,1,2)(0,1,2)[12] : Inf
## ARIMA(0,1,2)(1,1,0)[12] : 7056.93
## ARIMA(0,1,2)(1,1,1)[12] : 6967.235
## ARIMA(0,1,2)(1,1,2)[12] : 6956.061
## ARIMA(0,1,2)(2,1,0)[12] : 7049.062
## ARIMA(0,1,2)(2,1,1)[12] : 6943.594
## ARIMA(0,1,3)(0,1,0)[12] : 7068.331
## ARIMA(0,1,3)(0,1,1)[12] : 6955.453
## ARIMA(0,1,3)(0,1,2)[12] : Inf
## ARIMA(0,1,3)(1,1,0)[12] : 7057.802
## ARIMA(0,1,3)(1,1,1)[12] : 6967.187
## ARIMA(0,1,3)(2,1,0)[12] : 7049.636
## ARIMA(0,1,4)(0,1,0)[12] : 7058.872
## ARIMA(0,1,4)(0,1,1)[12] : 6937.713
## ARIMA(0,1,4)(1,1,0)[12] : 7044.188
## ARIMA(0,1,5)(0,1,0)[12] : 7040.125
## ARIMA(1,1,0)(0,1,0)[12] : 7141.841
## ARIMA(1,1,0)(0,1,1)[12] : Inf
## ARIMA(1,1,0)(0,1,2)[12] : Inf
## ARIMA(1,1,0)(1,1,0)[12] : 7134.709
## ARIMA(1,1,0)(1,1,1)[12] : 7035.774
## ARIMA(1,1,0)(1,1,2)[12] : Inf
## ARIMA(1,1,0)(2,1,0)[12] : 7118.449
## ARIMA(1,1,0)(2,1,1)[12] : 6997.593
## ARIMA(1,1,0)(2,1,2)[12] : 6979.979
## ARIMA(1,1,1)(0,1,0)[12] : 7070.422
## ARIMA(1,1,1)(0,1,1)[12] : 6957.612
## ARIMA(1,1,1)(0,1,2)[12] : Inf
## ARIMA(1,1,1)(1,1,0)[12] : 7058.611
## ARIMA(1,1,1)(1,1,1)[12] : 6966.577
## ARIMA(1,1,1)(1,1,2)[12] : 6955.293
## ARIMA(1,1,1)(2,1,0)[12] : 7050.613
## ARIMA(1,1,1)(2,1,1)[12] : 6958.465
## ARIMA(1,1,2)(0,1,0)[12] : 7045.333
## ARIMA(1,1,2)(0,1,1)[12] : 6958.381
## ARIMA(1,1,2)(0,1,2)[12] : Inf
## ARIMA(1,1,2)(1,1,0)[12] : 7058.896
## ARIMA(1,1,2)(1,1,1)[12] : 6966.876
## ARIMA(1,1,2)(2,1,0)[12] : 7052.659
## ARIMA(1,1,3)(0,1,0)[12] : 7043.631
## ARIMA(1,1,3)(0,1,1)[12] : 6940.16
## ARIMA(1,1,3)(1,1,0)[12] : 7029.352
## ARIMA(1,1,4)(0,1,0)[12] : 7042.933
## ARIMA(2,1,0)(0,1,0)[12] : 7078.83
## ARIMA(2,1,0)(0,1,1)[12] : 6932.109
## ARIMA(2,1,0)(0,1,2)[12] : Inf
## ARIMA(2,1,0)(1,1,0)[12] : 7039.756
## ARIMA(2,1,0)(1,1,1)[12] : 6949.61
## ARIMA(2,1,0)(1,1,2)[12] : 6942.982
## ARIMA(2,1,0)(2,1,0)[12] : 7030.294
## ARIMA(2,1,0)(2,1,1)[12] : 6955.388
## ARIMA(2,1,1)(0,1,0)[12] : 7071.88
## ARIMA(2,1,1)(0,1,1)[12] : 6926.495
## ARIMA(2,1,1)(0,1,2)[12] : Inf
## ARIMA(2,1,1)(1,1,0)[12] : 7038.397
## ARIMA(2,1,1)(1,1,1)[12] : 6946.29
## ARIMA(2,1,1)(2,1,0)[12] : 7027.478
## ARIMA(2,1,2)(0,1,0)[12] : 7043.649
## ARIMA(2,1,2)(0,1,1)[12] : 6926.132
## ARIMA(2,1,2)(1,1,0)[12] : 7029.284
## ARIMA(2,1,3)(0,1,0)[12] : 7045.449
## ARIMA(3,1,0)(0,1,0)[12] : 7081.8
## ARIMA(3,1,0)(0,1,1)[12] : 6930.677
## ARIMA(3,1,0)(0,1,2)[12] : Inf
## ARIMA(3,1,0)(1,1,0)[12] : 7040.976
## ARIMA(3,1,0)(1,1,1)[12] : 6952.285
## ARIMA(3,1,0)(2,1,0)[12] : 7030.012
## ARIMA(3,1,1)(0,1,0)[12] : 7067.075
## ARIMA(3,1,1)(0,1,1)[12] : 6927.343
## ARIMA(3,1,1)(1,1,0)[12] : 7040.14
## ARIMA(3,1,2)(0,1,0)[12] : 7045.375
## ARIMA(4,1,0)(0,1,0)[12] : 7049.838
## ARIMA(4,1,0)(0,1,1)[12] : 6921.383
## ARIMA(4,1,0)(1,1,0)[12] : 7027.863
## ARIMA(4,1,1)(0,1,0)[12] : 7045.11
## ARIMA(5,1,0)(0,1,0)[12] : 7045.571
##
## Now re-fitting the best model(s) without approximations...
##
##
##
##
## Best model: ARIMA(4,1,0)(0,1,1)[12]
auto_f <- forecast(auto_fit,h=60,level=c(99.5))
plot(auto_f)
plot(ts_birth, main="Actual data & forecasts",ylab="Birth records")
lines(auto_f$fitted,col='red')
plot(auto_f$residuals, xlab="Time", ylab="Residuals")
hist(auto_f$residuals, xlab="Residuals", main="Residuals histogram")
Acf (auto_f$residuals, lag.max = 48, main = "Residuals ACF")
plot(auto_f$residuals~naive_f$fitted, main = "Residual-Fitted values plot",ylab="Residuals",xlab="Fitted values")
plot(auto_f$residuals~naive_f$x, main = "Residual-Actual values plot",ylab="Residuals",xlab="Actual values")
mean_f <- meanf(ts_birth,60)
plot(mean_f$residuals, xlab="Time", ylab="Residuals")
hist(mean_f$residuals, xlab="Residuals", main="Residuals histogram")
plot(mean_f$residuals~mean_f$fitted, main = "Residual-Fitted values plot",ylab="Residuals",xlab="Fitted values")
plot(mean_f$residuals~mean_f$x, main = "Residual-Actual values plot",ylab="Residuals",xlab="Actual values")
Acf (mean_f$residuals, lag.max=48, main="Residuals ACF")
plot(mean_f)
stl_birth <- stl(ts_birth, s.window='periodic')
stl_forecast <- forecast(stl_birth, h=60)
plot(stl_forecast$residuals, xlab="Time", ylab="Residuals")
hist(stl_forecast$residuals, xlab="Residuals", main="Residuals histogram")
plot(stl_forecast$residuals~stl_forecast$fitted, main = "Residual-Fitted values plot",ylab="Residuals",xlab="Fitted values")
plot(stl_forecast$residuals~stl_forecast$x, main = "Residual-Actual values plot",ylab="Residuals",xlab="Actual values")
Acf (stl_forecast$residuals, lag.max=48,main="Residuals ACF")
plot(stl_forecast)
plot(ts_birth, main="Actual data & forecasts",ylab="Birth records")
lines(stl_forecast$fitted,col='red')
naive_ac <- accuracy(naive_f)
mean_ac <- accuracy(mean_f)
sma3_ac <- accuracy(sma_3,ts_birth)
sma6_ac <- accuracy(sma_6,ts_birth)
sma12_ac <- accuracy(sma_12,ts_birth)
ses_ac <- accuracy(ses$fitted,ts_birth)
HW_ac <- accuracy(HW$fitted,ts_birth)
stl_ac <- accuracy(stl_forecast$fitted,ts_birth)
arima_ac <- accuracy(auto_f)
naive_mape <- naive_ac['Training set', 'MAPE']
mean_mape <- mean_ac['Training set', 'MAPE']
sma3_mape <- sma3_ac['Test set', 'MAPE']
sma6_mape <- sma6_ac['Test set', 'MAPE']
sma12_mape <- sma12_ac['Test set', 'MAPE']
ses_mape <- ses_ac['Test set', 'MAPE']
HW_mape <- HW_ac['Test set', 'MAPE']
stl_mape <- stl_ac['Test set', 'MAPE']
arima_mape <- arima_ac['Training set','MAPE']
mape_table <- data.frame(
Method = c("Naive","Mean", "SMA3", "SMA6", "SMA12", "SES", "HW","STL","ARIMA"),
MAPE = c(naive_mape, mean_mape,sma3_mape, sma6_mape, sma12_mape, ses_mape, HW_mape,stl_mape,arima_mape),
stringsAsFactors = FALSE
)
mape_table
## Method MAPE
## 1 Naive 4.303580
## 2 Mean 4.507706
## 3 SMA3 2.487378
## 4 SMA6 3.719305
## 5 SMA12 3.640532
## 6 SES 3.795787
## 7 HW 1.294030
## 8 STL 1.236984
## 9 ARIMA 1.167318